Why has cross entropy become the classification standard loss function and not Kullbeck Leibler divergence?
When it comes to classification problem in machine learning, the cross entropy and KL divergence are equal. As already stated in the question, the general formula is this:

$$H(p, q) = H(p) + D_{KL}(p||q)$$

Where $p$ a “true” distribution and $q$ is an estimated distribution, $H(p, q)$ is the cross-entropy, $H(p)$ is the entropy and $D$ is the Kullback-Leibler divergence.

Note that in machine learning, $p$ is a one-hot representation of the ground-truth class, i.e., 

$$p = [0,..., 1, ..., 0]$$

which is basically a delta-function distribution. But the entropy of the delta function is zero, hence KL divergence simply equals the cross-entropy.

In fact, even if $H(p)$ wasn't $0$ (e.g., soft labels), it is fixed and has no contribution to the gradient. In terms of optimization, it's safe to simply remove it and optimize the Kullback-Leibler divergence.